Abstract
Factor analysis is one of the most important statistical tools for analyzing multivariate data (i.e., items) in the social sciences. An essential case is the comparison of multiple groups on a one-dimensional factor variable that can be interpreted as a summary of the items. The assumption of measurement invariance is a frequently employed assumption that enables the comparison of the factor variable across groups. This article discusses different estimation methods of the multiple-group one-dimensional factor model under violations of measurement invariance (i.e., measurement noninvariance). In detail, joint estimation, linking methods, and regularized estimation approaches are treated. It is argued that linking approaches and regularization approaches can be equivalent to joint estimation approaches if appropriate (robust) loss functions are employed. Each of the estimation approaches defines identification constraints of parameters that quantify violations of measurement invariance. We argue in the discussion section that the fitted multiple-group one-dimensional factor analysis will likely be misspecified due to the violation of measurement invariance. Hence, because there is always indeterminacy in determining group comparisons of the factor variable under noninvariance, the preference of particular fitting strategies such as partial invariance over alternatives is unjustified. In contrast, researchers purposely define fitting functions that minimize the extent of model misspecification due to the choice of a particular (robust) loss function.
Highlights
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We have argued that joint estimation, linking, and regularized maximum likelihood (ML) estimation in the tau-equivalent and the tau-congeneric model can provide similar if not identical estimates in the violation of MI if an appropriate loss function ρ in joint estimation or linking is used
The wisdom under applied researchers that partial invariance is necessary for determining group-mean comparisons [24] is unsound because it would imply that a particular loss function should always be preferred in practice
Summary
We assume a one-dimensional factor F in the tau-equivalent model [14]: Xi = νi + F + εi , Var(εi) = φ , where the residuals εi are uncorrelated. Note that it is assumed that there are equal loadings λ and equal residual variances θ, while item intercepts νi are item-specific parameters. We assume E(F) = 0 and ψ = Var(F) is estimated. Denote by I the I × I identity matrix and by 1 an I × 1 vector of ones. The covariance matrix Σ of the items X is represented by a model-implied covariance matrix Σ0. Note that the covariance matrix is parsimoniously represented by only two parameters. The mean vector μ = E(X) = ν is estimated without constraints
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